Question

a and \(\underline{b}\) are unit vector the angle between the vectors is \(\theta\). \(|\underline{\mathrm{a}}+\underline{\mathrm{b}}|>1\). Then (a) \(\theta=(\pi / 2)\) (b) \(\theta<(\pi / 3)\) (c) \(\theta>(2 \pi / 3)\) (d) \((\pi / 2)<\theta<(2 \pi / 3)\)

Short answer

The short answer is: (b) \(\theta<(\pi / 3)\).

Step-by-step solution

Write the Formula for the Magnitude of the Sum of Two Vectors

Using the formula for the magnitude of the sum of two vectors, we can write it as follows: \( |\underline{a} + \underline{b}| = \sqrt{|\underline{a}|^2 + |\underline{b}|^2 + 2 |\underline{a}| |\underline{b}| \cos \theta} \) Since both a and b are unit vectors, their magnitudes are equal to 1. So, we can simplify the equation: \( |\underline{a} + \underline{b}| = \sqrt{1^2 + 1^2 + 2(1)(1) \cos \theta} \)

Set the Inequality for the Magnitude of the Sum of Two Vectors

We are given that the magnitude of the sum of the two vectors should be greater than 1. So, we can write the inequality: \( \sqrt{1 + 1 + 2 \cos \theta} > 1 \) Squared both sides: \( 1 + 1 + 2 \cos \theta > 1^2 \)

Simplify the Inequality

Now we simplify the inequality and solve for θ: \( 2 + 2 \cos \theta > 1 \) \( 2 \cos \theta > -1 \) \( \cos \theta > -\frac{1}{2} \)

Find the Range of θ that Satisfies the Inequality

Since the cosine function is positive in the first and fourth quadrants and negative in the second and third quadrants, we can find the range of θ where the inequality is satisfied: In the first quadrant, we have: \( 0 < \theta < \frac{\pi}{3} \) In the second quadrant, the cosine is negative, but the inequality is not satisfied because \( \cos \theta \) is smaller than -1/2. In the third quadrant, the cosine is also negative, and again, the inequality is not satisfied. In the fourth quadrant, we have: \( \frac{5\pi}{3} < \theta < 2\pi \) So, combining both valid ranges, we get: \( 0 < \theta < \frac{\pi}{3} \) or \( \frac{5\pi}{3} < \theta < 2\pi \) Comparing with the given options, it is seen that (b) \(\theta<(\pi / 3)\) is true, but this isn't the complete answer, since it doesn't take into account the range between \(5\pi/3\) and \(2\pi\). However, none of the given options fully matches our calculation. The best choice based on the available information would be to select the closest matching option, which is (b) \(\theta<(\pi / 3)\).

Key concepts

Vector Algebra

Vector algebra is an essential topic in mathematics and physics that deals with operations involving vectors. Vectors are quantities that have both magnitude and direction, and they are used to represent physical entities like force, velocity, and displacement.

In vector algebra, we learn how to add and subtract vectors, as well as how to multiply them by a scalar. Vector addition is commutative, meaning that the sum does not change irrespective of the order of the vectors being added. When we have two vectors, say \( \underline{a} \) and \( \underline{b} \) which are added together, the resultant vector \( \underline{a} + \underline{b} \) points from the tail of the first vector to the tip of the second when they are placed head to tail.

One common operation in vector algebra is finding the magnitude of the sum of two vectors, which is a key aspect of this exercise. It utilizes the Pythagorean theorem in a two-dimensional plane, and it's extended into three dimensions using the dot product to account for angles between vectors.

Magnitude of Vectors

The magnitude of a vector is a measure of its 'length' and represents the distance from its starting point (or tail) to its endpoint (or head). It is denoted by vertical bars, for example, \( |\underline{a}| \). Magnitude is always a non-negative scalar, and it is calculated differently depending on the dimension of the vector.

For a two-dimensional vector with components \( x \) and \( y \) on the Cartesian plane, the magnitude is found using the formula \( |\underline{a}| = \sqrt{x^2 + y^2} \). In the context of our exercise, we are working with unit vectors, which have a magnitude of 1. This simplicity helps in calculations, but the concept of magnitude is foundational to solving problems involving vectors in physics and engineering, as it's used to represent the size of physical quantities.

Cosine Rule for Vectors

The cosine rule, also known as the law of cosines, is a formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. When applied to vectors, this rule helps us find the magnitude of the sum or difference of two vectors when the angle between them is known.

For vectors \( \underline{a} \) and \( \underline{b} \) at an angle \( \theta \), the cosine rule states that: \( |\underline{a} + \underline{b}|^2 = |\underline{a}|^2 + |\underline{b}|^2 + 2|\underline{a}||\underline{b}|\cos\theta \).

This equation played a pivotal role in solving our original exercise problem. By understanding how the cosine of an angle affects the magnitude of the sum of two vectors, one can deduce a range of possible values for that angle. Such applications of the cosine rule are prevalent in physics, especially when decomposing forces or velocities into their components.